Let us recall from, e.g., [1]} that a 2-cocycle on a Hopf algebra \(K\)  is
a convolution-invertible linear map \(\sigma \colon K\otimes K\rightarrow \mathbb {C}\) 
that is unital, i.e., \(\sigma (a\otimes \mathbf {1}) = \epsilon (a) = \sigma (\mathbf {1}\otimes a)\) 
for all \(a\in K\) , and that satisfies the cocycle condition
\(\sigma (b{_{\underline{1}}} \otimes c{_{\underline{1}}}) \sigma (a\otimes b{_{\underline{2}}} c{_{\underline{2}}})= \sigma (a{_{\underline{1}}} \otimes b{_{\underline{1}}}) \sigma (a{_{\underline{2}}} b{_{\underline{2}}}\otimes c) ,\) 
